Finite Group Databasebeta

← #14320852

Subgroups — #14320852

Order 1920.

1304 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D20C211yesmaximal, characteristic
2C2×#91393C212yesmaximal, complemented
3#11751073C211yesmaximal, characteristic, complemented
4C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#65277C211yesmaximal, characteristic
6Dic3×#685C212yesmaximal, complemented
7#11748841C211yesmaximal, characteristic, complemented
8#11748796C211yesmaximal, characteristic
9#11748708C212yesmaximal, complemented
10#11748707C212yesmaximal, complemented
11C4×Dic3×Dic5C211yesmaximal, characteristic, complemented
12#10605348—31nomaximal, Hall
13C22×C3×D20C2211yescharacteristic
14C22×D5×Dic3C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17C2×#6712C2212yes—
18C2×C4×Dic15C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2212yes—
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C3×#685C2212yes—
23C2×C3×#685C2212yes—
24C2×C3×C4×Dic5C2211yescharacteristic
25C2×#6669C2212yes—
26C2×#6665C2212yes—
27C2×#6665C2212yes—
28C2×#6665C2212yes—
29C2×Dic3×Dic5C2211yescharacteristic
30C2×Dic3×Dic5C2211yescharacteristic
31C2×Dic3×Dic5C2211yescharacteristic
32C2×Dic3×Dic5C2211yescharacteristic
33Dic3×D20C418yescomplemented
34#91393—28no—
35#91115C2212yes—
36C5×C3⋊#95C2212yes—
37#90994C2212yes—
38#76289—51nomaximal, Hall
39C2×C4×D20—31no—
40C2×#4018—32no—
41C4×C5×C4⋊C4—31no—
42#65277—31no—
43#65277D311yescharacteristic
44#65262—31no—
45#65262—31no—
46C4×#685—32no—
47C42×Dic5—31no—
48#64958—32no—
49#64958—32no—
50C23×C3×D5C2312yes—
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2311yescharacteristic
53C22×Dic15C2312yes—
54C22×Dic15C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2311yescharacteristic
58C22×C5×Dic3C2311yescharacteristic
59C22×C3×Dic5C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C2×C3×D20—22no—
62C2×C3×D20—22no—
63C2×C3×D20C2×C414yes—
64C2×D5×Dic3—24no—
65C2×D5×Dic3—24no—
66C2×D5×Dic3C2×C418yes—
67C3×C5×C4⋊C4—22no—
68C4⋊Dic15C2×C414yes—
69#6712—24no—
70C4×Dic15—22no—
71C5×C4⋊Dic3—22no—
72C4×C5×Dic3—24no—
73C4×C5×Dic3C2×C414yes—
74C3×#685—24no—
75C3×#685—24no—
76C3×C4×Dic5—22no—
77#6669—24no—
78#6665—24no—
79#6665—24no—
80#6665C2×C418yes—
81Dic3×Dic5—22no—
82Dic3×Dic5—22no—
83Dic3×Dic5—22no—
84Dic3×Dic5D414yescomplemented
85C2×D4×Dic3—51no—
86C2×#877—52no—
87C3×#390—51no—
88#5221—51no—
89Dic3×C4⋊C4D511yescharacteristic, complemented
90#4972—51no—
91#4945—52no—
92#4944—52no—
93Dic3×C22⋊C4—52no—
94C42×Dic3—51no—
95#4928—51no—
96C22×D20C2×D311yescharacteristic
97C22×C4×D5—32no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C42×C5—31no—
101C2×C42×C5—32no—
102C2×#685—32no—
103C2×#685—32no—
104C2×#685—32no—
105C2×#685C2×D312yes—
106C2×#685C2×D312yes—
107C2×C4⋊Dic5—31no—
108C2×#683—32no—
109C2×#683—32no—
110C2×C4×Dic5—31no—
111C2×C4×Dic5—31no—
112C2×C4×Dic5—31no—
113C2×C4×Dic5—31no—
114C2×C4×Dic5—31no—
115C2×C4×Dic5C2×D311yescharacteristic
116C4×D20—38no—
117#4018—68no—
118C5×#95—32no—
119#3963—32no—
120#3963—32no—
121#2293—151noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C3×D5—22no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—22no—
128C22×C3×D5—22no—
129C22×C3×D5C22×C414yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5#4112yes—
135C2×C3×C4×C5#4112yes—
136C2×Dic15—22no—
137C2×Dic15—21no—
138C2×Dic15—21no—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—22no—
142C2×Dic15C22×C414yes—
143C2×Dic15C2×D412yes—
144C2×Dic15#4112yes—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3C22×C412yes—
151C2×C5×Dic3C22×C412yes—
152C2×C5×Dic3C2×D412yes—
153C2×C5×Dic3#4112yes—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5C2×D412yes—
159C2×C3×Dic5#4112yes—
160C3×D20—28no—
161D5×Dic3—216no—
162C22×C3×D4—51no—
163C23×Dic3—52no—
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C22⋊C4—52no—
166C2×C3×C22⋊C4—52no—
167C2×C3×C42—51no—
168C2×#217—52no—
169C2×#217—52no—
170C2×#217—52no—
171D4×Dic3—58no—
172C2×C4⋊Dic3—51no—
173C2×C4⋊Dic3C2×D511yescharacteristic
174C2×Dic3⋊C4—52no—
175C2×Dic3⋊C4—52no—
176C2×C4×Dic3—51no—
177C2×C4×Dic3—51no—
178C2×C4×Dic3—51no—
179C2×C4×Dic3—51no—
180C2×C4×Dic3—51no—
181C2×C4×Dic3C2×D512yes—
182C2×C4×Dic3C2×D511yescharacteristic
183#877—108no—
184C3⋊#95—52no—
185C3⋊#95—52no—
186C3⋊#95C2×D512yes—
187C23×D5C22×D312yes—
188C22×C4×C5—31no—
189C22×C4×C5—31no—
190C22×C4×C5—32no—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C22×Dic5—32no—
194C22×Dic5—31no—
195C22×Dic5—31no—
196C22×Dic5C22×D311yescharacteristic
197C22×Dic5C22×D311yescharacteristic
198C2×D20—22no—
199C2×D20—22no—
200C2×D20C4×D314yes—
201C2×C4×D5—38no—
202C2×C4×D5—64no—
203C2×C4×D5—64no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4—62no—
206C42×C5—34no—
207C42×C5—64no—
208#685—24no—
209#685—24no—
210#685—38no—
211#685—64no—
212#685—64no—
213C4⋊Dic5—34no—
214#683—64no—
215#683—64no—
216C4×Dic5—22no—
217C4×Dic5—34no—
218C4×Dic5—62no—
219C4×Dic5—62no—
220C4×Dic5—62no—
221C4×Dic5—62no—
222C2×C4×D4—151no—
223C2×#117—152no—
224#390—51no—
225#390—151no—
226#389—151no—
227#389—151no—
228#388—152no—
229#388—152no—
230C4×C4⋊C4—31no—
231C4×C22⋊C4—152no—
232C43—151no—
233C22×C3×C5C23×C411yescharacteristic
234C22×C3×C5C22×D411yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5C2×#4111yescharacteristic
237C22×C3×C5C2×#4111yescharacteristic
238C22×C3×C5C2×#4111yescharacteristic
239C22×C3×C5C2×#4111yescharacteristic
240C2×C3×D5—24no—
241C2×C3×D5—24no—
242C2×C3×D5—24no—
243C2×C3×D5—42no—
244C2×C3×D5—42no—
245C2×C3×D5—42no—
246C2×C3×D5—42no—
247C3×C4×C5—42no—
248C3×C4×C5#11714yescomplemented
249Dic15—24no—
250Dic15—22no—
251Dic15—22no—
252C3×Dic5—22no—
253C3×Dic5—22no—
254C5×Dic3—22no—
255C5×Dic3—42no—
256C5×Dic3C4×D414yes—
257C24×C3—52no—
258C2×C3×D4—54no—
259C2×C3×D4—102no—
260C2×C3×D4—102no—
261C22×C3×C4—51no—
262C22×C3×C4—51no—
263C22×C3×C4C22×D512yes—
264C22×C3×C4C22×D511yescharacteristic
265C22×Dic3—51no—
266C22×Dic3—52no—
267C22×Dic3—51no—
268C22×Dic3—58no—
269C22×Dic3—104no—
270C22×Dic3—104no—
271C22×Dic3C22×D512yes—
272C22×Dic3C22×D511yescharacteristic
273C22×Dic3C22×D511yescharacteristic
274C3×C4⋊C4—22no—
275C3×C22⋊C4—104no—
276C3×C22⋊C4—104no—
277C3×C42—102no—
278#217—58no—
279#217—104no—
280#217—104no—
281C4⋊Dic3—22no—
282C4⋊Dic3—54no—
283Dic3⋊C4—104no—
284Dic3⋊C4—104no—
285C4×Dic3—24no—
286C4×Dic3—54no—
287C4×Dic3—102no—
288C4×Dic3—102no—
289C4×Dic3—102no—
290C4×Dic3—102no—
291C4×Dic3C4×D514yes—
292C23×C5C23×D311yescharacteristic
293C22×D5—22no—
294C22×D5—22no—
295C22×D5—22no—
296C22×D5—22no—
297C22×D5—22no—
298C22×D5—22no—
299C22×D5C2×C4×D314yes—
300C2×C4×C5—22no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—32no—
304C2×C4×C5—32no—
305C2×C4×C5—32no—
306C2×C4×C5—32no—
307C2×C4×C5—61no—
308C2×C4×C5—61no—
309C2×C4×C5—62no—
310C2×C4×C5—62no—
311C2×C4×C5—62no—
312C2×C4×C5C2×C4×D312yes—
313C2×C4×C5#23712yes—
314C2×C4×C5#23712yes—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—34no—
320C2×Dic5—32no—
321C2×Dic5—32no—
322C2×Dic5—62no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—62no—
328C2×Dic5D3×D412yes—
329C2×Dic5#23712yes—
330D20—28no—
331C4×D5—616no—
332C22×D4—51no—
333C23×C4—152no—
334C4×D4—158no—
335#117—308no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—151no—
338C2×C4⋊C4—152no—
339C2×C4⋊C4—152no—
340C2×C4⋊C4D3×D511yescharacteristic
341C2×C22⋊C4—52no—
342C2×C22⋊C4—52no—
343C2×C22⋊C4—152no—
344C2×C22⋊C4—152no—
345C2×C22⋊C4—152no—
346C2×C42—31no—
347C2×C42—32no—
348C2×C42—51no—
349C2×C42—151no—
350C2×C42—151no—
351C2×C42—151no—
352C2×C42—151no—
353C2×C42—151no—
354#95—32no—
355#95—152no—
356#95—152no—
357C2×C3×C5C2×#11711yescharacteristic
358C2×C3×C5C2×C4×D411yescharacteristic
359C2×C3×C5C4×#4111yescharacteristic
360C2×C3×C5#52911yescharacteristic
361C2×C3×C5#52911yescharacteristic
362C2×C3×C5#53211yescharacteristic
363C2×C3×C5#53211yescharacteristic
364C3×D5—44no—
365C23×C3—54no—
366C23×C3—102no—
367C23×C3—102no—
368C23×C3—102no—
369C23×C3—102no—
370C23×C3—102no—
371C23×C3—102no—
372C23×C3C23×D511yescharacteristic
373C3×D4—108no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—52no—
378C2×C3×C4—52no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4C2×C4×D512yes—
384C2×C3×C4#70912yes—
385C2×C3×C4#70912yes—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—54no—
392C2×Dic3—52no—
393C2×Dic3—52no—
394C2×Dic3—101no—
395C2×Dic3—101no—
396C2×Dic3—102no—
397C2×Dic3—101no—
398C2×Dic3—102no—
399C2×Dic3—101no—
400C2×Dic3—1016no—
401C2×Dic3C2×C4×D512yes—
402C2×Dic3C2×C4×D512yes—
403C2×Dic3D4×D512yes—
404C2×Dic3#71312yes—
405C22×C5C22×C4×D311yescharacteristic
406C22×C5C2×D3×D411yescharacteristic
407C22×C5C2×#23711yescharacteristic
408C22×C5C2×#23711yescharacteristic
409C22×C5C2×#23711yescharacteristic
410C22×C5D3×#4111yescharacteristic
411C22×C5D3×#4111yescharacteristic
412C2×D5—24no—
413C2×D5—24no—
414C2×D5—24no—
415C2×D5—42no—
416C2×D5—42no—
417C2×D5—42no—
418C2×D5—42no—
419C4×C5—34no—
420C4×C5—42no—
421C4×C5—62no—
422C4×C5—122no—
423C4×C5#87714yescomplemented
424Dic5—22no—
425Dic5—22no—
426Dic5—62no—
427Dic5—62no—
428Dic5—64no—
429C24—52no—
430C2×D4—54no—
431C2×D4—102no—
432C2×D4—102no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—32no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—158no—
439C22×C4—151no—
440C22×C4—152no—
441C22×C4—151no—
442C22×C4—304no—
443C22×C4—304no—
444C22×C4C2×D3×D512yes—
445C22×C4C2×D3×D511yescharacteristic
446C4⋊C4—22no—
447C4⋊C4—62no—
448C4⋊C4—154no—
449C4⋊C4—304no—
450C4⋊C4—304no—
451C22⋊C4—104no—
452C22⋊C4—104no—
453C22⋊C4—158no—
454C22⋊C4—304no—
455C22⋊C4—304no—
456C42—34no—
457C42—64no—
458C42—102no—
459C42—154no—
460C42—302no—
461C42—302no—
462C42—302no—
463C42—302no—
464C3×C5#229311yescharacteristic, complemented, Hall
465C22×C3—104no—
466C22×C3—104no—
467C22×C3—104no—
468C22×C3—202no—
469C22×C3—202no—
470C22×C3—202no—
471C22×C3—202no—
472C22×C3C22×C4×D511yescharacteristic
473C22×C3C2×#70911yescharacteristic
474C22×C3C2×#70911yescharacteristic
475C22×C3C2×D4×D511yescharacteristic
476C22×C3C2×#71311yescharacteristic
477C22×C3D5×#4111yescharacteristic
478C22×C3D5×#4111yescharacteristic
479C3×C4—42no—
480C3×C4—102no—
481C3×C4—102no—
482C3×C4#401814yescomplemented
483Dic3—22no—
484Dic3—42no—
485Dic3—104no—
486Dic3—102no—
487Dic3—102no—
488Dic3#403914yes—
489C2×C5C2×#87711yescharacteristic
490C2×C5C4×#23711yescharacteristic
491C2×C5C4×D3×D411yescharacteristic
492C2×C5#559611yescharacteristic
493C2×C5#560011yescharacteristic
494C2×C5#568011yescharacteristic
495C2×C5#571611yescharacteristic
496D5—44no—
497C23—54no—
498C23—102no—
499C23—102no—
500C23—102no—
501C23—102no—
502C23—102no—
503C23—102no—
504C23C22×D3×D511yescharacteristic
505D4—108no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—32no—
510C2×C4—32no—
511C2×C4—32no—
512C2×C4—32no—
513C2×C4—52no—
514C2×C4—52no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—154no—
525C2×C4—152no—
526C2×C4—152no—
527C2×C4—3016no—
528C2×C4—302no—
529C2×C4—301no—
530C2×C4—301no—
531C2×C4—302no—
532C2×C4—301no—
533C2×C4—301no—
534C2×C4#676512yes—
535C2×C4C4×D3×D512yes—
536C2×C4#677912yes—
537C2×C3—204no—
538C2×C3C2×#401811yescharacteristic
539C2×C3C2×#403911yescharacteristic
540C2×C3#6585611yescharacteristic
541C2×C3#6588211yescharacteristic
542C2×C3#6588211yescharacteristic
543C2×C3#6591411yescharacteristic
544C2×C3#6591411yescharacteristic
545C5#7628911yescharacteristic, complemented, Sylow
546C22—104no—
547C22—104no—
548C22—104no—
549C22—202no—
550C22—202no—
551C22—202no—
552C22—202no—
553C22C2×#676511yescharacteristic
554C22C2×C4×D3×D511yescharacteristic
555C22D3×D4×D511yescharacteristic
556C22D5×#23711yescharacteristic
557C22D3×#71311yescharacteristic
558C22#9203411yescharacteristic
559C22C2×#677911yescharacteristic
560C4—34no—
561C4—42no—
562C4—62no—
563C4—102no—
564C4—102no—
565C4—122no—
566C4—302no—
567C4—304no—
568C4—302no—
569C4#9139314yescomplemented
570C3#1060534811yescharacteristic, complemented, Sylow
571C2—204no—
572C2#1175339811yescharacteristic
573C2#1175341311yescharacteristic
574C2#1175349811yescharacteristic
575C2D3×#403911yescharacteristic
576C2C2×#9139311yescharacteristic
577C2#1175477711yescharacteristic
578C2#1175520411yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C4⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×D20C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2293
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#91393C2semidirect
3. #11751073C2semidirect
4. C5×Dic3×C4⋊C4C2semidirect
6. Dic3×#685C2semidirect
7. #11748841C2semidirect
9. #11748708C2semidirect
10. #11748707C2semidirect
11. C4×Dic3×Dic5C2semidirect
33. Dic3×D20C4semidirect
84. Dic3×Dic5D4semidirect
89. Dic3×C4⋊C4D5semidirect
248. C3×C4×C5#117semidirect
423. C4×C5#877semidirect
464. C3×C5#2293semidirect
482. C3×C4#4018semidirect
545. C5#76289semidirect
569. C4#91393semidirect
570. C3#10605348semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×Dic3×D202
C2×#913932
#117510732
C5×Dic3×C4⋊C42
C3×#652772
Dic3×#6852
#117488412
#117487962
#117487082
#117487072
C4×Dic3×Dic52
#106053483
#762895